Theorem 3.2 ([1]}, [2]}) 
Let \(T: (X, d)\rightarrow (X,d)\)  and \(S:Y\rightarrow Y\)  be two TDSs and \(\pi :X\rightarrow Y\)  a factor map. Then
for any \(\nu \in \mathcal {M}_S(Y)\)  and any potential \(\varphi \in C(X, \mathbb {R})\) ,
\(\begin{aligned}\sup _{\mu \in \mathcal {M}_T(X)} \Big \lbrace h_\mu (T, X|Y)+\int _X \varphi (x) d\mu (x): \pi \mu =\nu \Big \rbrace =\int _Y P(T, \varphi , y)d\nu (y).\end{aligned}\) 
